1.1 Textbook Learning Objectives

• Understand and use the terminology of probability.
• Determine whether two events are mutually exclusive and whether two events are independent.
• Calculate probabilities using the Addition Rules and Multiplication Rules.
• Construct and interpret Contingency Tables.
• Construct and interpret Venn Diagrams.
• Construct and interpret Tree Diagrams.

1.2 Instructor Learning Objectives

• Begin to appreciate how probability connects back to the broader idea of statistics
• Understand the emphasis of chance and probability, not certainty, in statistics

1.3 Introduction

• Most decisions we make are a (hopefully educated) guess.
  • E.g., Should I wear my boots if it might rain today?, What are the chances that my class will respond well to this activity?
  • Chance and probability is intuitive, and, if we introspect closely, it plays a role in pretty much everything we do

• Discuss: Try coming up with more examples of weighing probability in your own life

• Probability is a core part of statistics, and our results are often phrases in recognizing that level of uncertainty
  • This is why you never really hear good scientists say we have definitively proved something - there’s always room left for uncertainty
• There is a lot of vocabulary (and math) surrounding probability
  • You could take an entire semester long class just talking about probability and the philosophy/math of it
  • We’ll try to stay centered on exploring it as relevant to statistics, though this module may initially feel very removed from statistics

• Important: Even though we won't be 'in the weeds' as much with probability calculations later on, it will remain an important background subject!

2.1 Introduction

• Like with the last module, practice makes perfect
  • Please use the various exercises in the book and visit GA and/or professor office hours!
  • This is also a great time to embrace flashcards, if that historically works well for you

2.2 Rapid Fire Vocabulary

• In the context of probability, an experiment is some tightly controlled and monitored action - similar to how we use the term in research design
  • If the experiment is not pre-determined, that is, it’s outcome is not guaranteed, then there is some amount of chance
  • So if we call something a chance experiment, its some controlled action that we are not sure of the outcome
  • Classic example is a coin flip

• Question: Example of how probability is intuitive - what is the chance that I flip a coin and it lands on 'tails'
• A) 25\%
• B) 75\%
• C) 50\%
• D) 0\%

• The final result of the experiment is called an outcome
  • The possibilities of all the different outcomes is called the sample space (notation = \(S\))
  • Sample space is represented as a list, a tree, or Venn diagram; more example of these later
  • Coin flip with possible heads (\(H\))/tails (\(T\)) outcome example = \(S = \{H,T\}\)
• An event is some combination of outcome(s)
  • Important to know that it could just be one outcome or more than on described outcome
  • Events will be defined as upper case letters like \(A\), \(B\), \(C\), etc.
• The probability of a certain event occurring is described as the long-term relative frequency of said event
  • What that means is that the probability describes how often a certain event would occur if the experiment were to be completed an infinite number of times; this hinges on the law of large numbers
  • The probability of a certain event happening will be written as \(P(x)\), where \(x\) is whatever event is being looked at
    • \(P(x)\) will always be \(\geq 0\) and \(\leq 1\)
  • When \(P(x)\) of all possible outcomes is equal to one another, we would say that this experiment is equally likely, sometimes also said to be fair
• Example of coin toss, where event \(A\) is the likelihood of flipping with an outcome of head on a single toss, and \(B\) is the likelihood of tails
  • If \(P(A) = 0.50\), i.e., a 50% chance to land on heads
  • If \(P(B) = 0.50\), i.e., a 50% chance to land on tails
  • If \(P(A) = P(B)\), i.e., equally likely experiment

• Discuss: Try writing out the same set of bullets that I just did, but with a six-sided equally likely die instead. We will calculate it out in a second, but try to just do this intuitively now.

2.4 OR Events

• An OR Event is when an event is written in such a manner that an outcome could match event \(A\) or \(B\)
  • This would be written as \(A \cup B\) (that symbol is called a cup)
  • Applying the above, we would have a two step process of determining the outcome match \(A\) OR \(B\) and would give the probability as \(P(A \cup B)\)
• To list the possible outcomes, we find the set union of the two sets of outcomes that match the respective events, example:
  • If \(A\) matches \(\{1, 2\}\) and
  • If \(B\) matches \(\{2, 3, 4\}\) then
  • \(A \cup B\) matches \(\{1, 2, 3, 4\}\)
  • Then we proceed by using \(A \cup B\) matches in step 4 of Calculating Probability

2.5 AND Events

• An AND Event is when an event is written in a manner that an outcome could match event \(A\) or \(B\)
  • This would be written as \(A \cap B\) (this symbol is a cap, like a hat)
  • Thus probability would be written as \(P(A \cap B)\)
• To list the possible outcomes, we find the set intersection of the two sets of outcomes that match the respective events, example:
  • If \(A\) matches \(\{1, 2, 3\}\) and
  • If \(B\) matches \(\{2, 3, 4\}\) then
  • \(A \cap B\) matches \(\{2, 3\}\)
  • Then we proceed by using \(A \cap B\) matches in step 4 of Calculating Probability

• Important: A mnemonic to remember caps and cups: cAps are for Ands

2.6 Conditionals and Complements

• Complements take the inverse, or everything but the matching outcomes of an event
  • This is written at the prime of an outcome, e.g., \(A'\) is the complement of \(A\)
  • Thus, probability of a conditional would be notated as \(P(x')\)
  • In such a scenario, instead of finding all the outcomes that DO match the event, we would list all the outcomes that dont match the event in step 3 of Calculating Probability
• Conditionals are the probability a certain event occurs given that another event has already happened
  • It is written as \(P(A|B)\), which translates to probability of \(A\) given \(B\) (the vertical bar is sometimes called a pipe)
  • This is calculated via the following formula, assuming \(P(B) \neq 0\):

\[ P(A|B) = \frac{A \cap B}{B} \]

• Once again, try working through the problems and solutions in your book for all of these scenarios!

3.1 Introduction

• We are often interested in how two possible events are potentially related or unrelated
  • We can investigate both the independence and mutually exclusivity of events to determine this
  • These are both special description

3.2 Independence

• Two events are considered to be independent if they are unrelated/do not affect one another
  • On the contrary, if they are somehow related to one another, they are said to be dependent
• The following are the possible rules to show independence:
  • \(P(A|B) = P(A)\)
  • \(P(B|A) = P(B)\)
  • \(P(A \cap B) = P(A)P(B)\) (The “Multiplication Rule”)
  • Only one of these conditions needs to be met for establishing independence
• It is safest to assume dependence of events until independence has been established
  • Sort of counter-intuitive!

Connection to Sampling Process

• Recall that in a random sampling scenario, there is a theoretically equal chance for each member of the population of interest to be selected
  • Effectively, random sampling is sampling that is done equally likely
• Sampling can be done one of two ways, that impact whether sampling decisions were made independent of one another
  • Sampling can be done with replacement which means the same individual could be chosen multiple times - this means the odds of being selected as a member of the population remain the same
  • Sampling can also be done without replacement, meaning it is done in a manner that a person cannot be chosen multiple times, thus reducing the outcomes by one and changing the odds for each subsequent selection

3.3 Mutually Exclusivity

• Events are mutually exclusive when they cannot occur at the same time and follow the “Addition Rule”:

\[ P(A \cap B) = 0.00 \]

• If unsure, assume non-exclusivity until this rule is met

4.1 Introduction

• The word “contingent” means to depend on, so contingency tables offer a nice view for displaying probabilities in such a way to investigate dependence

• Important: Like with descriptive statistics, using graphical and tabular displays can make it much easier to navigate complex information, make sure you use them!

4.2 Example of a Contingency Table

5.1 Introduction

• Just like with how descriptive statistics have several ways to graphically present, so do probabilities!
  • Contingency tables are a great start, but we can add to them with tree diagrams and venn diagrams
• For this section, I’ll be a bit more brief only because these representations are not super practical in most applied statistics

5.2 Tree Diagrams

5.3 Venn Diagrams

6.1 Recap

• Probability is a tough subject, but it is critical as a foundation topic prior to understanding inferential statistics!

• Understand that when we discuss statistical results going forward, it will always be in through the lens of probabilities and likelihood of outcomes.

• While “real” statistics won’t often involve intensive hand calculation of probabilities, you should still have a foundation appreciation of how to do it - just like we did with standard deviation.

6.2 Lecture Check-in

• Make sure to complete any lecture check-in tasks associated with this lecture!

Module 3 Lecture - Transformations and Non-parametric Comparisons for Two Groups || Analysis of Variance